It follows easily that \(M_t Q_t \mid {\mathcal {F}}_{t-1}\)  is mean-zero and \(8\operatorname{lr}(1/\delta )*{M_t}_2 *{v_t}\sqrt{d}\) -norm-subGaussian. Hence, by [1]}, we know that there exists an absolute constant \(c > 0\)  such that for any \(\theta > 0\)  and \(\delta > 0\) , we have that with probability at least \(1 - \delta \) ,
\(*{\sum _{t=0}^{\tau -1} M_t Q_t} \le c \theta \sum _{t=0}^{\tau - 1} d(\operatorname{lr}(1/\delta ))^2 *{M_t}_2^2 *{v_t}^2 + \frac{1}{\theta } \log (2d/\delta ).\) 
